Ph.D. Qualifying Exam - University of Wisconsin–Milwaukeeuwm.edu/engineering/wp-content/uploads/sites/57/2014/03/me... · Refrigerant 134a enters an insulated compressor operating

MECHANICAL ENGINEERING DEPARTMENT

PhD Qualifying Exam

Part I

Mathematics and Fundamentals

Closed BookNotes

November 15 2007

Thursday

100 pm - 500 pm

Enter problem number(s)

The problems are that you selected

Ordinary Differential Equations 2 problems ( ) ( )

Partial Differential Equations 2 problems ( ) ( )

Linear Algebra 2 problems ( ) ( )

Calculus 2 problems ( ) ( )

Numerical Methods 2 problems ( ) ( )

The student should select 8 problems out of 10 problems given here Write the

problem numbers that you selected in ( ) above Also circle the problems that you

choose on each problem sheet Choose at least one problem from each category

listed above Use one exam book (blue book) for each problem Include your

assigned number and NOT your name on each book Submit both exam books and

this problem sheet when you leave

QualExam-F07Fall 2007

Ordinary Differential Equation 1

1) Find the solution y(t) of the initial value problem

0 (0) 2 (0) 3y y y yprimeprime primeprimeprime primeprimeprime primeprimeprime prime+ = = = minus+ = = = minus+ = = = minus+ = = = minus

2) What are the maximum and minimum values of the solution y(t) for minusinfin lt t lt infin

Ordinary Differential Equation 2

Consider a damped spring-mass system The position from equilibrium y at a

time t is described by the equation

0452

2

=++ ydt

dy

dt

yd

The momentum for the system is given by p = dydt

For the initial conditions of y(0) = 4 p(0) = -5 find

(a) the equation describing the position y as a function of time

(b) the equation describing the momentum p as a function of time and

(c) the total energy of the system at t = 3

Partial Differential Equation 1

The differential equation to describe the transient heat conduction is given as

0 1le lex

part

part

part

part

2

2

2

20

T

x

T

y+ =

with the boundary conditions of T as

BC

00 ==part

partxat

x

T

axatHTx

T==+

part

part0

0)( == yatxfT

byaty

T==

part

part0

where H is the convective heat transfer which you can treat as a constant

Obtain an expression for the steady state temperature distribution T(xy) in a rectangular region

byax lelelele 00

y

x a

b

0

Partial Differential Equation 2

Consider a semi-infinite rod of rectangular cross section (2Ltimes2 l ) as shown in the figure below

The base temperature of the rod is θ0 and the ambient temperature is 0 The heat transfer

coefficient is large Determine the steady state temperature of the rod using separation of

variables The governing equation can be described as follows

02

2

2

2

2

2

=part

part+

part

part+

part

part

zyx

θθθ

The boundary conditions are

0)(0)0(

==part

partzyL

x

zyθ

θ

0)(0)0(

==part

partzx

y

zxlθ

θ

)(0)()0( 0 finiteyxyx =infin= θθθ

Linear Algebra 1

Let F 4real 3real be the linear mapping defined by

F(x y s t) = (x ndash y + s + t x + 2s ndash t x + y + 3s ndash 3t)

Find a basis and the dimension of the image of U of F

Show all of your work

Linear Algebra 2

Determine the equation of the circle passing through the following three points in the plane (26) (64)

and (71)

Calculus 1

A fluid flows out of a region bounded by the curve 6x2 + 4 y

2 = 24 The velocity

field of the fluid is defined by

F(xy) = (3y + y2 + 8x) I + (x

4 ndash x + 6y ndash 3) j

Find the rate of flow out of the region The velocity is measured in cms and the

area is measured in cm2

Calculus 2

If )4(2 trnet

minus=θ what value of n will make

1 2

2 trr

rr part

part=

part

part

part

part θθ

Numerical Methods 1

Consider the typical diffusion equation

2

2

x

u

t

u

part

part=

part

part

Prove that when the following explicit scheme is used the spatial order of accuracy is second

order

2111 )21(h

krwhereruurruu jijijiji =+minus+= +minus+

Here h and k are spatial and temporal grid size What value of r should be assigned if 4th

order

spatial accuracy is required

Numerical Methods 2

Using the second-order terms of the Taylor series expansion and the forward-time and centered-

space differencing show that the advection equation

0=part

part+

part

part

x

fu

t

f (1)

can be formulated as

( )n

i

n

i

n

i

n

i

n

i

n

i

n

i fffc

ffc

ff 11

2

11

1 22

)(2

minus+minus++ +minus+minusminus=

where

x

tuc

∆

∆=

What is the order of the accuracy

Using von Neumann stability analysis determine the stability condition

MECHANICAL ENGINEERING DEPARTMENT

PhD Qualifying Exam

Part II

Area of Concentration

Thermal Science Stem

Open BookNotes

November 16 2007

Friday

100 pm ndash 500 pm

Enter problem number(s)

The problems are that you selected

Fluid Mechanics 3 problems ( ) ( ) ( )

Thermodynamics 3 problems ( ) ( ) ( )

Heat Transfer 3 problems ( ) ( ) ( )

The student should select 6 problems out of 9 problems given here Write the problem

numbers that you selected in ( ) above Also circle the problems that you choose on

each problem sheet Choose at least one problem from each category listed above

Use one exam book (blue book) for each problem Include your assigned number and

NOT your name on each book Submit both exam books and this problem sheet when

you leave

QualExam-F07Fall 2007

Fluid Mechanics 1

A cylindrical annular flow with the inner cylinder rotating at a constant speed ωωωω and

the outer cylinder stationary is shown in the figure below The radii of the outer and

inner cylinders are Ro and Ri respectively

(1) Express the velocity along the wall in terms of the dimensions rotating

speed=ωωωω and the fluid properties viscosity=micromicromicromicro and density=ρρρρ

(2) What is the power needed to move per unit depth Assume Ro ndash Ri ltlt Ro

(3) For SAE 30 oil at 20oC with Ro=10cm Ri=9cm and ωωωω=900rpm calculate

the power

[Hint] The equi-dimensional ordinary differential equation of the function fI

frdquo + (1r)frsquo ndash(1r2)f = 0

has a form of the general solution in the form of f1=C1r and f2=C2r

Fluid Mechanics 2

A cylindrical rod rests on a wall of 45deg A gate is mounted on the left top side of

the rod to hold water Compute the horizontal and vertical forces due to the water

acting on the 6-ft-diameter cylindrical rod per unit width (1-ft)

612rsquo

A B

C

D

088rdquo 45deg

Fluid Mechanics 3

A flow field is given by kyjzyixv ˆ)3(ˆ2ˆ2 22 +minusminus=r

bull Show that the flow field is irrotational

bull Derive an expression for the velocity potential Ф within an arbitrary

constant

bull If ξ is a rather complicated path from (0 0 0) to (0 0 4) use Ф to

compute the integral int bull= RdvIrr

over ξ where kzjyixR ˆˆˆ ++=r

is the

position vector of a point on the curve

bull What is the value of I if the integration is performed over a closed

path Do not repeat the calculations performed above to get to the

answer Hint use a well known theorem

Thermodynamics 1

Refrigerant 134a enters an insulated compressor operating at steady state as saturated

vapor at -10degC at a rate of 18 kgmin At the compressor exit the pressure of the

refrigerant is 5 bar Kinetic and potential energy effects can be neglected If the rate

of entropy production must be kept less than 04842 kJkmin determine the possible

range of values for isentropic compressor efficiency

Thermodynamics 2

A turbojet engine operates between pressure limits of 5 and 50 psia The inlet air

temperature to the compressor is -40degF and the upper temperature limit for the

engine is 2000degF Calculate the thrust for 1 lbmsec of air flow assuming

isentropic compression and expansion and an inlet velocity of 300 ftsec Assume

constant specific heats Also calculate the heat input per pound mass of air

Inlet

AIR

Exhaust

gas

Burner

QH

Thermodynamics 3

A liquid-vapor mixture of water with a mass of 03 kg at 150oC occupies an

insulated tank with a volume of 01 m3 Water vapor (and vapor alone) is removed

from the tank until the temperature is 140oC Determine the amount of water that is

removed during the process

Heat Transfer 1

The temperature of a gas stream is to be measured by a thermocouple whose junction

can be approximated as a 1-mm-diameter sphere The properties of the junction are

k=35 Wm-degC ρ=8500 kgm3 and Cp=320 Jkg-degC and the convection heat transfer

coefficient between the junction and the gas is h=210 Wm2-degC Determine how long

it will take for the thermocouple to read 99 of the initial temperature difference

Heat Transfer 2

A liquid metal flowing with uniform velocity V through a pipe of periphery P and

cross section A is electrically heated at the rate of u primeprimeprime over a length of l of the pipe

The inlet temperature of the liquid is equal to the ambient temperature Tinfin The

outside heat transfer coefficient is h Neglecting the effect of axial conduction and

you should use the lumped temperature across the cross-sectional area of the pipe

for part (a)

(a) Find the steady axial temperature distribution of the liquid metal

(b) The flow of the liquid metal is suddenly stopped because of a pump

failure Find the unsteady temperature of the liquid metal Hint treat

entire heating region as a single control volume

Heat Transfer 3

MECHANICAL ENGINEERING DEPARTMENT

PhD Qualifying Exam

Part II

Area of Concentration

Machine Design Stem

Open BookNotes

November 16 2007

Friday

100 pm ndash 500 pm

Enter problem number(s)

The problems are that you selected

Machine Design 3 problems ( ) ( ) ( )

Kinematics amp Dynamics 3 problems ( ) ( ) ( )

Controls amp Vibration 3 problems ( ) ( ) ( )

The student should select 6 problems out of 9 problems given here Write the problem

numbers that you selected in ( ) above Also circle the problems that you choose on

each problem sheet Choose at least one problem from each category listed above

Use one exam book (blue book) for each problem Include your student ID number

and NOT your name on each book Submit both exam books and this problem sheet

when you leave

QualExam-F07Fall 2007

Machine Design 1

The figure fellow shows a vehicular jack which is to be designed to support a maximum load of 5000

Newtons The link angle θ varies from 15 to 70 deg The links are 300 mm long Using an assumed

FOS and a suitable cross-section for the links determine the required standard size for the links

Machine Design 2

The figure below shows portion of a pump that is gear driven at uniform load and speed The shaft is

made of AISI 1045 cold-drawn steel and is supported by bearings mounted in the pump housing

The forces acting on the gear are shown below Find the factor of safety with respect to fatigue

failure at the fillet

Machine Design 3

For the C-clamp shown below the operator can comfortably exert a 20 lb force at the end of the

handle which creates a clamping force of W = 1000 lb The screw has American Standard

Threads (2θ = 60deg) with 13 threads per inch (single-threaded) The outside diameter is 050 inch

The root diameter and root area are 04001 inch and 01257 inch2 respectively Knowing that dc

= 050 inch micro t = 012 and microc = 025 determine the required handle length L

Kinematics amp Dynamics 1

The rotations of an airplane about a set of xyz axes attached to the airplane are termed the roll

rate wx the pitching rate wy and the yaw rate wz A passenger on a jumbo jet 30 m behind and 2

m above the center of mass G of the airplane is walking at a constant rate of 1 msec Determine

the difference between the acceleration that this passenger experiences and the acceleration that

the center of mass of the airplane undergoes if the roll rate is 05 radsec the pitching rate is -03

radsec and the yaw rate is 01 radsec All rotation rates are constant

Show all of your work

Kinematics amp Dynamics 2

The two bars are released from rest at the position θ

Determine their angular velocities at the instant they

become horizontal Neglect the mass of the roller at C

Each bar has a mass m and length L

θ θ

A

B

C

L L

Kinematics amp Dynamics 3

For the 1-DOF mechanism shown below

(a) Derive the loop equations and identify the known and unknowns in these equations

(b) Derive the velocity equations and identify the known and unknowns in these equations

(c) Derive the acceleration equations Identify the known and unknowns in these equations

Controls amp Vibration 1

Controls amp Vibration 2

Consider the following closed-loop control system

1) Determine the system type with respect to the disturbance input D(s)

2) Determine the sensitivity of the closed-loop transfer function T(s) from D(s) to Y(s) to

changes in the parameter A ie ( )T s

AS

3) Let D(s) = 0 determine the steady state error eSS for a unit ramp reference input ie R(s)

2

1( )R s

s====

4) Let D(s) = 0 determine the sensitivity of the steady state error eSS for unit ramp reference

input with respect to changes in K ie SSe

AS

R(s) Y(s)

+ minus

( 1)

A

s sτ ++++ K

+

+

D(s)

Controls amp Vibration 3

Find the natural frequencies and modes shapes of the vibrating system shown in the figure below

The following data applies m1 = 1 kg m2 = 2 kg m3 = 3 kg k1 = 300 Nm k2 = 500 Nm k3 =

200 Nm 1 = 075 m